Permission to Write Apex Code

 They need the Author Apex permission. Ticking this box will result in them being assigned a lot of other permissions as well though.

Granting Author Apex permission also gives the user the following permissions:

• Modify All Data
• View Setup and Configuration

Granting Modify All Data also gives the user the following permissions:

Read, Create, Edit, Delete, View All, Modify All on all standard and custom objects

• Edit Events
• Edit Tasks
• Manage Public List Views
• Manage Public Templates
• Run Reports
• Transfer Record
• View Setup and Configuration
• Manage Public Documents
• Import Leads
• Transfer Leads
• View All Data
• Use Team Reassignment Wizards
• Manage Categories
• Convert Leads
• Import Solutions
• Create and Set Up Communities
• Connect Organization to Environment Hub
• Manage Reports in Public Folders
• Manage Dashboards in Public Folders
• Delete Topics
• Assign Topics
• Create Topics
• Edit Topics